1. For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point. We call the equation of the tangent the linearization of the function.

3. The linearization is the equation of the tangent line, and you can use the old formulas if you like. Start with the point/slope equation: linearization of f at a is the standard linear approximation of f at a.

4. Linearization

7. Example Finding a Linearization

8. Important linearizations for x near zero: This formula also leads to non-linear approximations:

9. Differentials: When we first started to talk about derivatives, we said that becomes when the change in x and change in y become very small. dy can be considered a very small change in y. dx can be considered a very small change in x.

10. Estimating Change with Differentials

11. Let be a differentiable function. The differential is an independent variable. The differential is:

12. Example Finding the Differential dy

13. Examples Find dy if a.y = x 5 + 37x Ans: dy = (5x 4 + 37) dx b.y = sin 3x Ans: dy = (3 cos 3x) dx

14. Differential Estimate of Change Let f(x) be differentiable at x = a. The approximate change in the value of f when x changes from a to a + dx is df = f ‘(a) dx.

15. Example The radius r of a circle increases from a = 10 m to 10.1 m. Use dA to estimate the increase in circle’s area A. Compare this to the true change ΔA.

16. Example: Consider a circle of radius 10. If the radius increases by 0.1, approximately how much will the area change? very small change in A very small change in r (approximate change in area)

17. Compare to actual change: New area: Old area: